MinT - Best Known (162−31, 162, s)-Nets in Base 4

Best Known (162−31, 162, s)-Nets in Base 4

(162−31, 162, 1118)-Net over F₄ — Constructive and digital

Digital (131, 162, 1118)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (23, 38, 90)-net over F₄, using
  - trace code for nets [i] based on digital (4, 19, 45)-net over F₁₆, using
    - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
        a function field by GarcÃa and Quoos [i]
2. digital (93, 124, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 31, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(162−31, 162, 8537)-Net over F₄ — Digital

Digital (131, 162, 8537)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶², 8537, F₄, 31) (dual of [8537, 8375, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁶², 16384, F₄, 31) (dual of [16384, 16222, 32]-code), using
  - an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(162−31, 162, 6205190)-Net in Base 4 — Upper bound on s

There is no (131, 162, 6205191)-net in base 4, because

1 times m-reduction [i] would yield (131, 161, 6205191)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 543958 229151 396968 789271 771064 233665 883002 450358 302785 835055 494971 967921 549045 345917 946762 352512 > 4¹⁶¹ [i]